Let's solve each of the given limits one by one.

11. $\lim_{{x \to 3}} (2x^2 + 4x + 1)$

We can directly substitute $x = 3$ into the quadratic expression: $2(3)^2 + 4(3) + 1 = 2(9) + 12 + 1 = 18 + 12 + 1 = 31$ So, the limit is $\boxed{31}$.

12. $\lim_{{x \to 1}} (3x^3 - 2x^2 + 4)$

Direct substitution of $x = 1$: $3(1)^3 - 2(1)^2 + 4 = 3 - 2 + 4 = 5$ The limit is $\boxed{5}$.

13. $\lim_{{x \to 3}} \sqrt{x + 1}$

Direct substitution of $x = 3$: $\sqrt{3 + 1} = \sqrt{4} = 2$ The limit is $\boxed{2}$.

14. $\lim_{{x \to 4}} \frac{3}{x + 4}$

Substitute $x = 4$: $\frac{3}{4 + 4} = \frac{3}{8}$ The limit is $\boxed{\frac{3}{8}}$.

15. $\lim_{{x \to 4}} (x + 3)^2$

Substitute $x = 4$: $(4 + 3)^2 = 7^2 = 49$ The limit is $\boxed{49}$.

16. $\lim_{{x \to 0}} (2x - 1)^3$

Substitute $x = 0$: $(2(0) - 1)^3 = (-1)^3 = -1$ The limit is $\boxed{-1}$.

17. $\lim_{{x \to 2}} \frac{1}{x}$

Substitute $x = 2$: $\frac{1}{2}$ The limit is $\boxed{\frac{1}{2}}$.

18. $\lim_{{x \to -3}} \frac{2}{x + 2}$

Substitute $x = -3$: $\frac{2}{-3 + 2} = \frac{2}{-1} = -2$ The limit is $\boxed{-2}$.

19. $\lim_{{x \to 1}} \frac{x}{x^2 + 4}$

Substitute $x = 1$: $\frac{1}{1^2 + 4} = \frac{1}{1 + 4} = \frac{1}{5}$ The limit is $\boxed{\frac{1}{5}}$.

20. $\lim_{{x \to -2}} \frac{2x - 3}{x + 5}$

Substitute $x = -2$: $\frac{2(-2) - 3}{-2 + 5} = \frac{-4 - 3}{3} = \frac{-7}{3}$ The limit is $\boxed{-\frac{7}{3}}$.

21. $\lim_{{x \to 7}} \frac{3x}{\sqrt{x + 2}}$

Substitute $x = 7$: $\frac{3(7)}{\sqrt{7 + 2}} = \frac{21}{\sqrt{9}} = \frac{21}{3} = 7$ The limit is $\boxed{7}$.

22. $\lim_{{x \to 2}} \frac{\sqrt{x + 2}}{x - 4}$

Substitute $x = 2$: $\frac{\sqrt{2 + 2}}{2 - 4} = \frac{\sqrt{4}}{-2} = \frac{2}{-2} = -1$ The limit is $\boxed{-1}$.

Would you like a more detailed explanation for any of these solutions or have any specific questions?

Here are some related questions you can explore:

1. What happens if the limit approaches a point where the function is undefined?
2. How can we compute limits at infinity for rational functions?
3. What are the techniques for solving indeterminate forms like $\frac{0}{0}$?
4. How do you apply L'Hopital's Rule to limits?
5. What is the concept of continuity in relation to limits?

Tip: Always attempt to substitute the point into the limit first to see if direct substitution works. If not, explore other techniques like factoring or L'Hopital's Rule.